Maths

Elin

I have no clue how to graph for i. Really, I searched my book, my lecture notes, the internet. I'm turning to you, please tell me how to graph for imaginary numbers.

The problem I'm presenting is not the actual problem I'm doing, but it's similar.

f(x)=(x+2)^2(x-3)(x-(2+i))(x-(2-i))

I can graph fine until I get the the imaginary set. I don't even know how to punch it into a graphing calculator to try and figure it out that way. In fact, I don't own a graphing calculator. My college doesn't allow their use on tests so I always figured it was a waste of cash to buy one.

Merouan

Notice the two complex roots- are the roots complex conjugates? ie, are the roots (x-(a+bi))*(x-(a-bi)) = (x -a)^2 + b^2. This second form is one you can use to graph in a graphing calculator pretty easily.

Since they are complex conjugates we can obtain the form (x-2)^2 + 1. This is completely real. Is the problem to graph the whole expression by hand?

Elin

Yes, I have to graph the whole expression by hand. I've done most of the work to get the above form, usually the easy part for me is slapping it in the graph but for the life of me I don't remember him ever showing us how to graph the complex stuff. All that I really have left is graphing it and then proving the graph by determining the end behaviors at the zero's. I'm a biology major, I hate hate hate all the maths.

Merouan

I'm not sure how you go about graphing things usually (I don't know how much math you know), but if you notice, I showed you how to change the two complex roots into real terms... (basically, multiply out the two complex roots and you'll find that the coefficients are all real) So you can now plug it in a graphing calculator at least (technically you could have before, you just have to know how).

The thing is, although this function looks like it'll spit out complex numbers, it wont, because the two complex roots are complex conjugates of each other (meaning they are related to each other by multiplying the imaginary part of one root by -1. this gives you the other root). Explicitly, you get
f(x) = (x-2)^2 (x-3) (x^2 -4x +5).

How do you usually graph functions? I'm not sure how you are expected to graph polynomial functions with purely real roots even... if what I gave you before isn't enough to help you, can you tell us how you usually graph them? Do you look for the roots and plug in numbers around it to find the behavior, or do you know calculus?